Question

Find the directional derivative of f(x,y)=x^2y ^3+2x^4 y at the point (−3,−2) in the direction θ=2π/3. The gradient of f is: ∇f(x,y)= ∇f(−3,−2)= The directional derivative is:

Answer 1

The directional derivative of f(x,y)=x^2y^3+2x^4y at the point (-3,-2) in the direction θ=2π/3 is 54 + 24√3.

Step-by-step explanation:

To find the directional derivative of the function f(x,y) = x^2y^3 + 2x^4y at the point (-3,-2) in the direction θ = 2π/3, we need to find the gradient of f and then use the directional derivative formula. The gradient of f is given by ∇f(x,y) = (∂f/∂x, ∂f/∂y) = (2xy^3 + 8x^3y, 3x^2y^2 + 2x^4). Evaluating the gradient at (-3,-2), we get ∇f(-3,-2) = (-108, -48). Next, we use the directional derivative formula: D_θf = ∇f · u, where u is the unit vector in the direction θ. In this case, u = (cos(2π/3), sin(2π/3)) = (-1/2, √3/2). Evaluating the dot product, we have D_θf = (-108, -48) · (-1/2, √3/2) = 54 + 24√3.